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Calculus

Overview

  • Credit value: 15 credits at Level 4
  • Convenor: Dan McVeagh
  • Assessment: a short-problem set (20%), online quiz (10%) and two-hour examination (70%)

Module description

In this module we explore the fundamental concepts of differentiation and integration, as well as methods of approximation and real-world applications. Key topics include differentiation rules, integration techniques, numerical methods, and using calculus to analyse functions.

Indicative syllabus

  • Differentiation: derivatives of standard functions, the chain rule, the product rule, the quotient rule, the inverse function rule, implicit differentiation, logarithmic differentiation
  • Integration: integrals of standard functions, definite integration and the area under a curve, integration by substitution, integration by parts, integration of rational functions
  • Methods of approximation: the bisection method, the Newton-Raphson method, the Trapezium rule, Simpson’s rule, Maclaurin and Taylor approximations, power series of standard functions
  • Applications of calculus: tangents, stationary points, maxima, minima and points of inflexion, curve sketching, rates of change, motion in a straight line, arc length, volumes of revolution, first order ODEs: variables separable and integrating factors

Learning objectives

By the end of this module you will be able to:

  • demonstrate knowledge of the methods of differentiation and integration, and differentiate and integrate functions of one variable
  • demonstrate knowledge of the basic notation and terminology of calculus
  • demonstrate knowledge of, and use, simple numerical methods for solving equations and for evaluating definite integrals
  • express a function of one variable as a power series, and use this as an approximation for the function
  • demonstrate awareness of the importance of convergence to the solution of a problem when using a numerical method, and the fact that, in some cases, such a method may fail to produce a valid solution
  • use a mathematical computer package to investigate and find solutions to problems considered in the module.